fuzzy chemistry - an axiomatic theory for general chemistry · pa,gf,dn (bicocca) fuzzy chemistry...

Fuzzy ChemistryAn Axiomatic Theory for General Chemistry

P. Amato(1), G. F. Cerofolini(2), and D. Narducci(2)

(1): DISCO(2): Scienze dei Materiali

January 8, 2009

PA,GF,DN (Bicocca) Fuzzy Chemistry January 8, 2009 1 / 56

Introduction

Outline

1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)

2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework

3 The axiomsStick-and-ballMolecular graphs

4 The natural interpretation5 Examples6 Conclusions


Introduction The matter and its description

The Physical World: Hierarchical organization



Explanation and reduction

Different scientific theories are designed for explaining differentlevels of matterEach theory specializes in explaining its own particular set ofphenomenaObject studied at one level are composed of objects studied atlower levels

Does this mean that, in principle, the lowest level theory can subsumeal the higher-level science?



Reductionism and Emergence

the properties of a level are the statistical properties of the lowerlevel: hierarchythere exist properties which cannot be derived as statisticalproperties of the lower level: holism



The Physical World: Hierarchical description


Introduction General Chemistry (GC)

General chemistry: the missing formal theory

chemistry may be viewed as a special chapter of physics devotedto the description of atomic assemblies (molecules, radicals,. . . )in terms of their constituting nuclei and electrons in mutualinteractionIt is generally believed that a description of molecules in terms ofproperties of their atomic constituents (General Chemistry (GC))is impossible.



GC basic fact

molecules are described as networks of atoms linked by bonds:stick-and-ball modelThough GC speaks using terms taken from physics (like atomicmass, bond length and energy, or formal charge, just to name afew), the properties of its fundamental characters (atoms andbonds) are autonomously postulated rather than being derivedfrom physicsGC has long reach and large predictive power, even using thelimited mathematical apparatus of college algebra



Relevance of GC formalization

Metatheoreticalto remove an anomaly in the hierarchy of description

PracticalIn its history, modern Chemistry went through two ages:

1 before quantum mechanics: concepts and rules defined bydescription and induction, based on empirical observation

2 after quantum mechanics: reductionistic approach, chemicalproperties explained by deduction from general principles.

Still Chemistry retains most of the concept that were developed beforeBohr’s atoms



Atoms in the hierarchical description of matter

Fortunately, many important properties of objects containing verymany atoms-such as the boiling and freezing of water-can be obtainedfrom simplified models of the structure of atoms and the lawsgoverning their interactions1.

1J. L. Lebowitz, Statistical mechanics: A selective review of two central issues, Rev.Mod. Phys. 71, 2 (1999) centenary issue



What is an atom in a molecule?

The derivation of the Hirshfeld atoms in molecules from informationtheory is clarified. The importance for chemistry of the concept ofatoms in molecules (AIM) is stressed, and it is argued that thisconcept, while highly useful, constitutes a noumenon in the sense ofKant2.

2Robert G. Parr, Paul W. Ayers, and Roman F. Nalewajski, What Is an Atom in aMolecule? J. Phys. Chem. A, 109 (17), 3957 -3959 ( 2005)



Non-reductionist Chemistry: Formal chemistry (FC)

Aimto recover and to dignify these concepts, which are still in their infancyfrom the viewpoint of their formalization.

ApproachTo develop an axiomatic theory Formal chemistry (FC): combining

Formal logicGraph theoryFuzzy sets theory



Hierarchical description: FC

atoms

molecules

phases

systems

Statistical Mechanics

Thermodynanics

Formal Chemistry


Formal chemistry (FC): Underlying ideas

Outline






Formal chemistry (FC): Underlying ideas

Underlying ideas

(to some extent) a molecule may be regarded as a graphthe intrinsical uncertainty of GC can be included in a rigoroustheory by using characters taken from fuzzy logic

introduction of a “loose metric” for the description of atomic andmolecular properties

Chemical reactions preserve atoms but not bonds


Formal chemistry (FC): Underlying ideas Molecules and graphs

Molecules as graphs

atoms: verticesbonds: edges

Figure: The molecule CH4 as graph (V ,E)

Note: V = {C,H,H,H,H}, and E = {C−H,C−H,C−H,C−H}. V and Eare bags (or multisets).



A molecules is more than a graph

A molecule, however, is in general much more complicated thana graph

vertex: intrinsic physical properties (like mass, self-energy, radius,electronegativity and group of belonging)edge: geometric and energetic properties (bond length and bondenergy)each vertex has a formal charge governed by the number ofbonds afferent to it



Example: H2O molecule

Figure: Water molecule: space-filled model (left), dimensions and geometricstructure (right)

Source: Water. (2008, December 20). In Wikipedia, The Free Encyclopedia. Retrieved 08:56, December 24, 2008, from

http://en.wikipedia.org/w/index.php?title=Water&oldid=259124804


Formal chemistry (FC): Underlying ideas Loose metric

A (classical) probabilistic view is not suitable for thefoundation of general chemistry: an example

Example:character of theory: bond length l(a,b)

observable: internuclear distance d(a,b)

IssueIn general

|l(a,b)− d(a,b)| � measure error

Probabilistic solutiondefine l(a,b) as: E [all possible d(a,b)]but what “all possible” does mean?



A fuzzy view is suitable for the foundation of generalchemistry

Rather than thinking of the characters of FC as real numbers, it issufficient to think of them as fuzzy numbers3

3after quantum mechanics we should be ready to accept any even morecomplicate assignation, provided it works.



Fuzzy as a high theory

Fuzzy theory allows the separation axiom scheme of set theory tobe extended to predicates to which a precise truth value cannot beassignedIn this view, fuzzy theory is certainly a “high” theory allowing theextension of classical logic to fuzzy logicIn spite of that, however, fuzzy theory has had really few highapplications, its use being usually limited to cases where theuncertainty is due to a contingent poor knowledge of the situationunder analysis (think, for instance, of the fuzzy control of awashing machine).


Formal chemistry (FC): Underlying ideas Reactions

Chemical reactions

Ultimate goal of chemistry:succeeding in elucidating notonly the properties ofmolecules, but also theirevolutionChemical evolution isconstrained by conservation ofmatter: the transformation of asubstance into another impliesthe consumption of the formerand the formation of the latter

Figure: The reactionCH4 + 2 O2 −−→ CO2 + 2 H2O

image source: http://www.chemistryland.com/


Formal chemistry (FC): Underlying ideas Reactions

Linear Logic

Chemical evolution (the state transition associated with chemicalreaction) and finite resources can be tackled by using the features oflinear logic operators:

CH4 ⊗ O2 ⊗ O2 ( CO2 ⊗ H2O⊗ H2O,

where, use ⊗ (times) is the linear conjunction and ( (“lollipop”) thelinear implication


Formal chemistry (FC): Underlying ideas Logical Framework

FC theory: formal framework

SyntaxMany-sorted linear predicate logic

Two sortsGraphs (sort GR)Numbers (sort NU)

Linear logiclinear conjunction ⊗ (times)linear implication ( (“lollipop”).

SemanticsNatural interpretation

(sub)set of elements of sort GR: moleculeselements of sort NU: fuzzy numbers


The axioms

Outline






The axioms

Symbols

Besides the usual logical symbols4 (equality, conjunction,. . . ), FCrequires the following non-logical symbols5

N constant symbols c1, . . . , cN of sort GR (balls).The functional symbols e, l ,m, r , ξ, ζ,E ,M (GR 7→NU)The functional symbols Φ (NU 7→NU)The binary functional symbols e, l (GR×GR 7→NU)The predicate symbols Mol and ! (sort GR)All the needed constants (0,1) and functional symbols (+, . . .) forsort NU

4Here we consider linear logic as an extension of classical logic, and in this paperwe use both the symbols of usual and linear logics (each with its meaning).

5The concepts of graph, vertex, link, distance and angle are taken frommathematics and do not need to be defined in formal chemistry.


The axioms Stick-and-ball

The sort GR

Definition (sort GR)Any 3-dimensional connected graph τ = (V ,E) where:

1 V is a bag of balls (constants of sort GR)2 each vertex is labeled by the value z ∈ N, referred to as formal

charge.3 each edge is labeled by the value n ∈ N, referred to as multiplicity.

is an element of sort GR.NOTE: from now on τ = (B(τ),S(τ)).

Definition (Sticks)A stick is an edge (unordered couple) defined on constants of sort GR

(balls).



Representing Balls



Representing Balls: a 3D view



Axiom (Four-dimensional ordering of balls)

The balls (costants of sort GR) of formal chemistry are arranged in a4-dimensional ordering. This order relation is described by quadruples(P,G,D,F )a of integer numbers with the following domains:

P = 1, · · · ,7 :P = 1 ⇒ D = 0 ∧ F = 0 ∧G = 1,8

P = 2,3 ⇒ D = 0 ∧ F = 0 ∧G = 1, · · · ,8

P = 4,5 ⇒{

F = 0 ∧ D = 0 =⇒ G = 1, · · · ,8F = 0 ∧G = 0⇒ D = 1, · · · ,10

P = 6,7 =⇒

(F = 0 ∧ D = 0)⇒ G = 1, · · · ,8F = 0 ∧G = 0⇒ D = 1, · · · ,10D = 2 ∧G = 0⇒ F = 1, · · · ,14

aIn chemistry the variable P is known as period, the variables G and D are knownas main group and transition group, respectively. The quadruples with G = 0 arecalled transition atoms; those with P = 6 and F 6= 0 form the lantanide series whereasthose with P = 7 and and F 6= 0 form the actinide series.



Definition (“Ballic” number)

Consider an arrangement of balls (constants of sort GR) in order ofincreasing P, inside each P of increasing G, with the insertion ofconstants with the ordering imposed by D in between G = 2 andG = 3, with the ordering imposed by F in between D = 1 and D = 2 forP = 6 and of the constants with the ordering imposed by F in betweenD = 1 and D = 2 for P = 7. The natural number giving the ordinal ofeach ball in this sequence is called the ballica number Z .

a“Ballic” is a neologism coined by the authors. Its meaning will be clarified atsemantic level.



4D ordering and ballic numbers

Figure: The 4-dimensional order (the P,G,D,F outside the table) and theballic numbers (the numbers inside the table) of the constants of sort GR,where N = 118



Balls: Physical properties

Axiom (Balls)

The balls are hard massive three-dimensional spheres, with radius(with functional symbol r ), mass (with functional symbol m), selfenergy(with functional symbol e) and electronegativity (with functional symbolξ) such that:

∀b (r(b) > 0 ∧ m(b) > 0 ∧ e(b) ≥ 0 ∧ ξ(b) > 0).



Sticks

Since each stick s is uniquely identified by a couple {a,b} of balls, inprinciple the functions on sticks take two arguments. The followingaxiom aims at reducing these functions to functions with just oneargument

Axiom (Sticks)

Let s = {a,b}

l(s) = r(a) + r(b) (stick length)e(s) = e(a) + e(b) + Φ(|ξ(a)− ξ(b)|) (stick energy)


The axioms Molecular graphs

Imposing constraints on graphs

Graps have topological propertiesIn FC the metric properties do matter.To introduce the notion of molecular graph we need to define twokind of properties (topological and metric) that the graph termshave to satisfy.



Definition (Topologically sound)A graph term is topologically sound if for any vertex b = (P,G,D,F )(ball) the following relationships are satisfied:

G − z + n − 8 = 0 if D = 0,F = 0 and G ≥ 3 (octet rule)G − z − n = 0 if D = 0,F = 0 and G ≤ 3 (group-valency rule)

where n denote the number of sticks (each counted for its multiplicity)connected to b, and z its formal charge.

Figure: Circles are balls and lines are sticks. The number iz inside a circleindicates the ball with ballic number i and formal charge z. The number overan edge is its multiplicity.



Definition (Metrically sound)A graph term τ is metrically sound if there exist a 3D spatialarrangement such that:

for any stick {a,b} in S(τ) the distance between the centers of thespheres a and b is l(a,b)

for any couple of balls a,b in B(τ), the intersection of the spheresa and b is null.

Figure: A topologically and metrically sound graph. r(b1) = 25 pm,r(b8) = 60 pm



The graphs relevant for formal chemistry

Axiom (Molecular graphs: predicate Mol)For any graph term τ , the predicate Mol(τ) is true if and only if τ istopologically and metrically sound. Each graph term τ such thatMol(τ) is true, is referred to as molecular graph. The set of allmolecular graphs, i.e. {τ : Mol(τ)}, is referred to as Γ set



Axiom (Molecular graph properties)

For all γ ∈ Γ

M(γ) =∑

b∈B(γ)

m(b) (m.g. mass)

ζ(γ) =∑

b∈B(γ)

zb (m.g. charge)

E(γ) =∑

s∈S(γ)

e(s), (m.g. energy)

where γ = (B(γ),S(γ)) and zb is the formal charge of the ball b.



Axiom (Allowed transition (!) predicate)Let L,R be bags of molecular graphs

L ! R ⇐⇒

∑γ∈L

B(γ) =∑γ′∈R

B(γ′) ∧∑γ∈L

ζ(γ) =∑γ′∈R

ζ(γ′)

The predicate {γ1, γ2}! γ3 is true when the “chemical” reaction thattransforms γ1 and γ2 in γ3 is possible. In the model this property isusually stated that in a chemical reaction matter (the bags of balls B)and charge (the formal charges zb) are conserved. On the contrary, ingeneral, bonds are not conserved in chemical reaction.



Figure: Example of allowed transition: H2 + F2 −−→ 2 HF.



Definition (Energy change (∆E))

Let L,R be bags of molecular graphs such that L ! R. Then:

∆E ,∑γ′∈R

E(γ′)−∑γ∈L

E(γ).

Axiom (Molecular graphs dynamics)

Let L,R be bags of molecular graphs such that F ! F ′

∆E > 0 =⇒

⊗γ∈L

React(γ) (⊗γ′∈R

Form(γ′)


The natural interpretation

Outline







Models for FC

A model U for the two-sorted language of FC is an ordered triple〈G ,N , I〉, where G and N are nonempty domains, and I is andinterpretation function with domain the set of all constants, functionsand relations of the language such that:

if b is a ball (constant symbol of sort GR), then I(b) ∈ G

if n is a constant symbol of sort NU, then I(n) ∈ N

if f (δ=1...δm) is an m-placed function symbol, where δi ∈ {GR,NU},then I(f ) is an m-placed function whose i-th argument belongs toG (if δi =GR) or to N (if δi =NU)if R(δ1...δn) is an n-placed relation symbol, where δi ∈ {GR,NU},then I(R) is an n-placed relation whose i-th argument belongs toG (if δi =GR) or to N (if δi =NU)



Natural interpretation

G is the set of adducts (i.e., any combination of bound atoms)the molecular graphs (the terms of sort GR satisfying the predicateM) are interpreted as moleculesN is the set F(R) of fuzzy numbers defined on Rthe symbols +, · · · , . . . are interpreted as the usual operators offuzzy arithmetic



F(R) as N

Any function φ :GR 7→NU is interpreted as a function that associatesfuzzy quantities with molecules:

I(φ) : B 7→ N .

The shape of the membership function describing the fuzzy numberscan be different for different functions. In particular for some functionalsymbol (like m) it boils down to singleton characteristic function (i.e. tocrisp numbers).



Summary of natural interpretation of FC theory

Syntactic object Natural Interpretation NoteBall Atom 3D sphereStick Bond Tangent spheres

Ballic formal charge Formal charge Integer, crispStick multiplicity Bond multiplicity Natural, crispBallic number Atomic number Natural, crisp

Mol “To be molecule” Crisp predicateMolecular graph Molecule 3D graph

m,M mass Real, crispr Atomics radius Real, fuzzyl internuclear distance Real, fuzzy

e,E Energy Real, fuzzyξ electronegativity Real, fuzzy

∆E Reaction enthalpy Real, fuzzy( Chemical reaction


Examples

Outline






Examples

Example: heat generated by a chemical reaction

Chemical reaction:

H2 + F2 −−→ 2 HF. (1)

atoms involved: H (hydrogen) and F (fluorine)Balls: H = (1,1,0,0) and F = (2,7,0,0)

AtomAtomicnumber

Atomicweigth(g/mole)

Atomicradius(pm)

Elec-tronega-tivity

Self-energy(kJ/mole)

H 1 1.007 25 2.20 218F 9 18.998 50 3.98 79

Table: Properties of the atoms. The reported values are used as centers ofthe membership functions.


Examples

Fuzzy radius and fuzzy bond length

Figure: Gaussian fuzzy numbers for atomic radius (up) and bond length(down).


Examples

Bond properties

Bond Bond energy (kJ/mole)Bondlength (pm)

H–H 436 50F–F 158 100H–F 297 + 100(2.2− 3.98)2 = 613.84 75

Table: Properties of the bonds. The reported values are the centers of themembership functions.


Examples

Reaction enthalpies

Bond Bond en-ergy (kJ/mole)

Number ofbonds

Right (+) H–F 613.84 2 1227.7Left (−) H–H 436 1 −436

F–F 157.8 1 −158Total 633.68

Substance Heat of re-action (kJ/mole)

Number ofsubstances

Right (+) HF 269 2 538Left (−) H2 0 1 0

F2 0 1 0Total 538

Table: Comparison of the reaction enthalpies for hydrogen fluoride, calculatedfrom bond energies (down) and from the heats of reaction (down)


Examples

Figure: Gaussian fuzzy numbers for self-energy (up) bond energy (center)and energy change (down).


Conclusions

Outline






Conclusions

Conclusions

To construct a formal theory of GC is possibleThis theory represents the reality only if its mathematical structureis based on fuzzy arithmeticsHave we really formulated a genuine theory of general chemistry?Of course no—the theory is both incomplete and inconsistent withsome experimental factsHowever, FC has two great advantages

it provides a rigorous axiomatic foundation for topics so dispersedthat can hardly be classified as a theory.it allows a description of matter where the properties of moleculesare given in terms of those of the constituting atoms and how theyare bonded


Conclusions

Next steps

to deeply investigate fuzzy and interval arithmeticto set-up the machinery for fuzzy interpretation of FCto define operative tools based on FC to analyze molecules andreactionsTo develop algorithms on graphs to investigate charge flow insidemolecules


fuzzy chemistry - an axiomatic theory for general chemistry · pa,gf,dn (bicocca) fuzzy chemistry...

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